Re: construct an epimorphism

For a in and n in Z, map (a, n) to n. That's trivial.

For what do you mean by " " and " "? My first thought was that " " was the set of permutations on 3 objects and " " was the set of pairs of complex numbers but in that case, is finite (containing 6 members) while is infinite so there cannot be such an epimorphism.

Re: construct an epimorphism

You haven't given enough information. The morphism is an epimorphism for the first one. The morphism is an epimorphism for the second one. I am not sure what more you are trying to do with this. If you want, you can use which would also be an epimorphism for the first one.

Re: construct an epimorphism

Originally Posted by HallsofIvy

For a in and n in Z, map (a, n) to n. That's trivial.

For what do you mean by " " and " "? My first thought was that " " was the set of permutations on 3 objects and " " was the set of pairs of complex numbers but in that case, is finite (containing 6 members) while is infinite so there cannot be such an epimorphism.

Typically, is the cyclic group of order , but I imagine that is the group the OP meant.

Re: construct an epimorphism

Originally Posted by HallsofIvy

For a in and n in Z, map (a, n) to n. That's trivial.

For what do you mean by " " and " "? My first thought was that " " was the set of permutations on 3 objects and " " was the set of pairs of complex numbers but in that case, is finite (containing 6 members) while is infinite so there cannot be such an epimorphism.